Postprocessing Variables
Several specialized variables specific to acoustics are predefined in the Acoustics Module and can be used when analyzing the results of an acoustic simulation. The variables are available from the expression selection menus when plotting.
In this section:
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Intensity Variables
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Power Dissipation Variables
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Pressure Acoustics, Boundary Mode Variables
In the COMSOL Multiphysics Reference Manual:
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Results Analysis and Plots
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Operators, Functions, and Constants
Intensity Variables
The propagation of an acoustic wave is associated with a flow of energy in the direction of the wave motion, the intensity vector I. The sound intensity in a specific direction (through a specific boundary) is defined as the time average of (sound) power per unit area in the direction of the normal to that area.
Knowledge of the intensity is important when characterizing the strength of a sound source — that is, the power emitted by the source. The power is given by the integral of n·I on a surface surrounding the source, where n is the surface normal. The intensity is also important when characterizing transmission phenomena, for example, when determining transmission loss or insertion loss curves.
The acoustic intensity vector I (SI unit: W/m2) is defined as the time average of the instantaneous rate of energy transfer per unit area (sound power) pu, such that
where p is the pressure and u the particle velocity. In the frequency domain (harmonic time dependence), the integral reduces to
(2-3)
where denotes complex conjugation. In the frequency domain the velocity is readily expressed in terms of the pressure as . Using the characteristic specific impedance for plane waves, the intensity can also be expressed in terms of the RMS pressure as
In the Pressure Acoustics, Frequency Domain interface the more general formulation given in Equation 2-3 is used to define the intensity.
For time-dependent problems, the equivalent quantity is the instantaneous value of the intensity, defined as
This expression is difficult to recover in Pressure Acoustics and would require the solution of an additional PDE to calculate the velocity from the pressure dependent variable. Only the intensity I (time averaged) is available as results and analysis variable in the frequency domain and can be selected from the expressions menus when plotting. The instantaneous intensity does exists as a postprocessing variable in transient interfaces such as Linearized Euler and Linearized Navier-Stokes where the velocity is solved for explicitly, see Modeling with the Aeroacoustics Branch for details.
The variables are defined in Table 2-3, Table 2-4, and Table 2-5. In the variable names, phys_id represents the interface name, for example, acpr for a Pressure Acoustics, Frequency Domain interface.
Table 2-3: Intensity Variables in 3D
Variable
Description
phys_id.I_mag
Magnitude of the intensity vector (frequency domain only)
phys_id.Ix
x-component of the intensity vector (frequency domain only)
phys_id.Iy
y-component of the intensity vector (frequency domain only)
phys_id.Iz
z-component of the intensity vector (frequency domain only)
Table 2-4: Intensity Variables in 2D Axisymmetric
Variable
Description
phys_id.I_mag
Magnitude of the intensity vector (frequency domain only)
phys_id.Ir
r-component of the intensity vector (frequency domain only)
phys_id.Iz
z-component of the intensity vector (frequency domain only)
Table 2-5: Intensity Variables in 2D
Variable
Description
phys_id.I_mag
Magnitude of the intensity vector (frequency domain only)
phys_id.Ix
x-component of the intensity vector (frequency domain only)
phys_id.Iy
y-component of the intensity vector (frequency domain only)
In the COMSOL Multiphysics Reference Manual:
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Results Analysis and Plots
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Expressions and Predefined Quantities
Power Dissipation Variables
Common to the Pressure Acoustics fluid models (also porous materials) and The Thermoviscous Acoustics, Frequency Domain Interface is that all the interfaces model some energy dissipation process, which stem from viscous and thermal dissipation processes. The amount of dissipated energy can be of interest as a results analysis variable or as a source term for a multiphysics problem. An example could be to determine the amount of heating in the human tissue when using ultrasound. In the Acoustics Module special variables exist for the dissipation.
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For the case of a plane wave propagating in the bulk of a fluid (the general thermal and viscous fluid models described in Thermally Conducting and Viscous Fluid Model) the dissipation is
and in the frequency domain after averaging over one period
(2-4)
where * in Equation 2-4 is the complex conjugate operator.
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In addition, an approximate expression for the dissipated energy density from a propagating plane wave exists for the Narrow Region Acoustics, the Poroacoustics models and attenuation in Pressure Acoustics. This total dissipated power density Qpw is defined by
where is the magnitude of the intensity vector I, and k is the wave number. This expression is an approximation and is only valid for traveling plane waves (or waves that are close to plane); however, it has many uses as a first estimate of the dissipation since it is easy to calculate in many different situations. The expression is, for example, not valid for standing waves in resonant systems. When the above expression is not valid, the dissipated energy should be calculated using an energy balance approach.
The power dissipation variables are defined in Table 2-6. In the variable names, phys_id represents the name (acpr, for example, for a pressure acoustics interface).
Table 2-6: Power dissipation Variables
Variable
Description
phys_id.diss_therm
Thermal power dissipation density
phys_id.diss_visc
Viscous power dissipation density
phys_id.diss_tot
Total thermal and viscous power dissipation density
phys_id.Q_pw
Plane-wave total dissipated power density
Pressure Acoustics, Boundary Mode Variables
A series of special variables exist for postprocessing after solving a boundary mode acoustics problem. They include in-plane and out-of-plane components of the velocity v and acceleration a.
The in-plane (ip) and out-of-plane (op) components to the acceleration and velocity are defined as
where n is the normal to the surface being modeled. The velocity and acceleration are defined in terms of the gradient of the pressure p as follows
and
where kn is the out-of-plane wave number solved for, m is a possible radial wave mode number, and is the tangential derivative along the boundary.
The boundary mode acoustics variables are defined in Table 2-7. In the variable names, phys_id represents the name (acbm, for example, for a Boundary Mode Acoustics interface).
Table 2-7: Boundary Mode Acoustics variables in 3D
Variable
Description
phys_id.vipx
In-plane velocity, x-component
phys_id.vipy
In-plane velocity, y-component
phys_id.vipz
In-plane velocity, z-component
phys_id.vip_rms
In-plane velocity RMS value
phys_id.aipx
In-plane acceleration, x-component
phys_id.aipy
In-plane acceleration, y-component
phys_id.aipz
In-plane acceleration, z-component
phys_id.aip_rms
In-plane acceleration RMS value
phys_id.vopx
Out-of-plane velocity, x-component
phys_id.vopy
Out-of-plane velocity, y-component
phys_id.vopz
Out-of-plane velocity, z-component
phys_id.vop_rms
Out-of-plane velocity RMS value
phys_id.aopx
Out-of-plane acceleration, x-component
phys_id.aopy
Out-of-plane acceleration, y-component
phys_id.aopz
Out-of-plane acceleration, z-component
phys_id.aop_rms
Out-of-plane acceleration RMS value